Mean Field Theory

What You Will Learn

• The Mean Field Approximation (MFA) simplifies complex systems by replacing fluctuating variables with their average values.

• When applied to functions of many interacting variables, MFA assumes that correlations can be neglected:

\[ \langle f(x_1, x_2, x_3, \dots) \rangle \approx f(\langle x_1 \rangle, \langle x_2 \rangle, \langle x_3 \rangle, \dots) \]

• This approximation is foundational across many areas of physics, including:

  • Electronic structure theory (e.g., Hartree–Fock)

  • Liquid state theory (e.g., van der Waals)

  • Condensed matter physics (e.g., Ising and Heisenberg models)

Application of Mean-Field Approximation (MFA) to the Ising Model

• In the Ising model, the mean-field approximation (MFA) decouples spin–spin interactions, reducing the system to a single macroscopic parameter: the average magnetization per spin, denoted by \( m \).

\[ \langle s_i s_j \rangle \approx \langle s_i \rangle \langle s_j \rangle = m^2 \]

\[ \langle H \rangle \approx -J \sum_{\langle ij \rangle} \langle s_i \rangle \langle s_j \rangle - h \sum_i \langle s_i \rangle = -\frac{1}{2} J q m^2 - h m \]

• The factor of \( \frac{1}{2} \) avoids double-counting spin pairs, and \( q \) is the coordination number—for instance, \( q = 4 \) for a 2D square lattice and \( q = 6 \) for a 3D cubic lattice.

Derivation of the Mean-Field Hamiltonian

• Consider the 2D Ising model where each spin \( s_i \) interacts with its \( q \) nearest neighbors:

\[ H_i = -J \sum_{\delta} s_i s_{i+\delta} - h s_i = -\left( J \sum_{\delta} s_{i+\delta} + h \right) s_i \]

• Define the local mean field at site \( i \) as:

\[ H_i^{\text{MF}} = J \sum_{\delta} s_{i+\delta} + h \]

• Under the mean-field approximation, replace neighboring spins with their average value:

\[ H_i^{\text{MF}} \approx J q m + h \]

• The total Hamiltonian then becomes:

\[ H \approx - (J q m + h) \sum_i s_i \]

• Taking the thermal average and accounting for double counting leads to the previously derived mean-field expression for \( \langle H \rangle \).

Self-Consistent Equation for Magnetization

• Partition Function: After applying MFA, the system becomes effectively non-interacting, and the total partition function factorizes as \( Z = z^N \), where \( z \) is the partition function of a single spin. For \( s_i = \pm 1 \):

\[ E(s_i = \pm 1) = \mp (J q m + h) \]

\[ z = e^{\beta (J q m + h)} + e^{-\beta (J q m + h)} \]

• The probability that a spin is in state \( s_i \) is:

\[ p(s_i) = \frac{e^{\beta s_i (J q m + h)}}{z} \]

• Average Magnetization: The mean value of a spin is then:

\[ m = \langle s_i \rangle = \sum_{s_i} s_i \, p(s_i) = \frac{e^{\beta (J q m + h)} - e^{-\beta (J q m + h)}}{e^{\beta (J q m + h)} + e^{-\beta (J q m + h)}} \]

\[ m = \tanh(\beta (J q m + h)) \]

• This self-consistent equation relates the average magnetization \( m \) to itself via the hyperbolic tangent function, capturing the thermal competition between spin alignment and disorder.

Application of Mean-Field Approximation (MFA) to the Ising Model

• In the Ising model, the mean-field approximation (MFA) decouples spin–spin interactions, reducing the system to a single macroscopic parameter: the average magnetization per spin, denoted by \( m \).

\[ \langle s_i s_j \rangle \approx \langle s_i \rangle \langle s_j \rangle = m^2 \]

\[ \langle H \rangle \approx -J \sum_{\langle ij \rangle} \langle s_i \rangle \langle s_j \rangle - h \sum_i \langle s_i \rangle = -\frac{1}{2} J q m^2 - h m \]

• The factor of \( \frac{1}{2} \) avoids double-counting spin pairs, and \( q \) is the coordination number—for instance, \( q = 4 \) for a 2D square lattice and \( q = 6 \) for a 3D cubic lattice.

Derivation of the Mean-Field Hamiltonian

• Consider the 2D Ising model where each spin \( s_i \) interacts with its \( q \) nearest neighbors:

\[ H_i = -J \sum_{\delta} s_i s_{i+\delta} - h s_i = -\left( J \sum_{\delta} s_{i+\delta} + h \right) s_i \]

• Define the local mean field at site \( i \) as:

\[ H_i^{\text{MF}} = J \sum_{\delta} s_{i+\delta} + h \]

• Under the mean-field approximation, replace neighboring spins with their average value:

\[ H_i^{\text{MF}} \approx J q m + h \]

• The total Hamiltonian then becomes:

\[ H \approx - (J q m + h) \sum_i s_i \]

• Taking the thermal average and accounting for double counting leads to the previously derived mean-field expression for \( \langle H \rangle \).

Self-Consistent Equation for Magnetization

• Partition Function: After applying MFA, the system becomes effectively non-interacting, and the total partition function factorizes as \( Z = z^N \), where \( z \) is the partition function of a single spin. For \( s_i = \pm 1 \):

\[ E(s_i = \pm 1) = \mp (J q m + h) \]

\[ z = e^{\beta (J q m + h)} + e^{-\beta (J q m + h)} \]

• The probability that a spin is in state \( s_i \) is:

\[ p(s_i) = \frac{e^{\beta s_i (J q m + h)}}{z} \]

• Average Magnetization: The mean value of a spin is then:

\[ m = \langle s_i \rangle = \sum_{s_i} s_i \, p(s_i) = \frac{e^{\beta (J q m + h)} - e^{-\beta (J q m + h)}}{e^{\beta (J q m + h)} + e^{-\beta (J q m + h)}} \]

\[ m = \tanh(\beta (J q m + h)) \]

• This self-consistent equation relates the average magnetization \( m \) to itself via the hyperbolic tangent function, capturing the thermal competition between spin alignment and disorder.

Free Energy of Mean-Field Ising Models

Defining the Macrostate \( M \)

• Consider a spin lattice of \( N \) sites, where each spin can point up or down. The total magnetization is given by:

\[ N = N_{+} + N_{-}, \qquad M = N_{+} - N_{-} \]

• Defining the magnetization per spin \( m = M/N \), the probabilities of spin-up and spin-down states are:

\[ p_{\pm} = \frac{1 \pm m}{2} \]

Entropy, Energy, and Free Energy

• Entropy is given by the Shannon expression using the probabilities of spin states:

\[ S(m) = -N k_B \left[ \left( \frac{1+m}{2} \right) \log \left( \frac{1+m}{2} \right) + \left( \frac{1-m}{2} \right) \log \left( \frac{1-m}{2} \right) \right] \]

• Energy in the mean-field approximation is the sum over spins in the presence of an average field:

\[ E(m) = -N \left[ \frac{Jq}{2} m^2 + h m \right] \]

• Free Energy is the usual Legendre transform:

\[ F(m) = E(m) - T S(m) \]

• Dimensionless Free Energy per Spin (useful for analysis and plotting):

\[ f(m) = \frac{F}{N k_B T} = - \left[ \frac{Jq}{2 k_B T} m^2 + \frac{h}{k_B T} m \right] + \left[ \left( \frac{1+m}{2} \right) \log \left( \frac{1+m}{2} \right) + \left( \frac{1-m}{2} \right) \log \left( \frac{1-m}{2} \right) \right] \]

Finding the Critical Temperature \( T_c \)

• The critical point is determined by the second derivative of the free energy:

\[ \frac{\partial^2 f}{\partial m^2} \bigg|_{m=0} = 1 - \frac{Jq}{k_B T} = 0 \]

\[ \Rightarrow \quad T_c = \frac{Jq}{k_B} \]

Magnetization as a Function of Temperature

• The equilibrium magnetization minimizes the free energy, leading to a self-consistency condition:

\[ m = \tanh \left( \frac{Jq}{k_B T} m + \frac{h}{k_B T} \right) \]

• Rearranged as:

\[ - \left( \frac{Jq}{k_B T} \right) m - \frac{h}{k_B T} + \tanh^{-1}(m) = 0 \]

• For zero external field (\( h = 0 \)), the equation simplifies to:

\[ m = \tanh \left( \frac{T_c}{T} m \right) \]

• For small \( m \), expand \( \tanh^{-1}(m) \approx m + \frac{1}{3} m^3 \), yielding:

\[ - \frac{T_c}{T} m + m + \frac{1}{3} m^3 = 0 \]

• Solving this, we find the magnetization near the critical point behaves as:

\[ m = \pm \sqrt{ \frac{3(T_c - T)}{T_c} } \]

Investigating various limits

The \(h=0\) MFA case

The equation can be solved in a self-consistent manner or graphically by finding intersection between:

• \(m =tanh(x)\)

• \(x = \frac{Jqm}{k_BT}\)

When the slope is equal to one it provides a dividing line between two behaviours.

\[k_B T_c =qJ\]

\[m = tanh \Big(\frac{Tc}{T} m \Big)\]

MFA shows phase transitio for 1D Ising model at finite \(T=T_c\)!

import ipywidgets as widgets
from ipywidgets import interactive, interact
import matplotlib.pyplot as plt
import numpy as np

# Constants
J = 1  # Interaction strength
k = 1  # Boltzmann constant (set to 1 for simplicity)

def entropy(m):
    # Handle log of zero by adding a small number to the argument
    small_number = 1e-10
    return -(1+m)/2 * np.log((1+m)/2 + small_number) - (1-m)/2 * np.log((1-m)/2 + small_number)

def free_energy(m, T):
    # Calculate the free energy for given m and T
    return -0.5 * J * m**2 - T * entropy(m)

def plot_free_energy(T):
    # Magnetization range
    m_values = np.linspace(-1, 1, 100)
    F_values = [free_energy(m, T) for m in m_values]
    
    # Plotting
    plt.figure(figsize=(8, 5))
    plt.plot(m_values, F_values, label=f'T = {T}')
    plt.xlabel('Magnetization (m)')
    plt.ylabel('Free Energy per Spin (F)')
    plt.title('Mean-field Free Energy vs. Magnetization')
    plt.legend()
    plt.grid(True)
    plt.show()

interactive(plot_free_energy, T=(0.1, 2, 0.1 ))

def mfa_ising_Tc(T=1, Tc=1):

    x  = np.linspace(-3,3,1000)
    
    f = lambda x: (T/Tc)*x
    m = lambda x: np.tanh(x)
    
    plt.plot(x,m(x), lw=3, alpha=0.9, color='green')
    
    plt.plot(x,f(x),'--',color='black')
    idx = np.argwhere(np.diff(np.sign(m(x) - f(x))))
    plt.plot(x[idx], f(x)[idx], 'ro')
    
    plt.legend(['m=tanh(x)', 'x'])
    
    plt.ylim(-2,2)
    plt.grid('True')
    plt.xlabel('m',fontsize=16)
    plt.ylabel(r'$tanh (\frac{Tc}{T} m )$')
    plt.show()

interact(mfa_ising_Tc, T=(0.1,5)) 

<function __main__.mfa_ising_Tc(T=1, Tc=1)>

def mfa_ising_h_vs_m(T=1):

    Tc = 1
    x  = np.linspace(-1,1,200)
    
    h = T*(np.arctanh(x) - (Tc/T)*x)
    
    plt.plot(x, h, lw=3, alpha=0.9, color='green')

    plt.plot(x, np.zeros_like(x), lw=1, color='black')
    plt.plot(np.zeros_like(x), x, lw=1, color='black')
     
    plt.grid(True)

    plt.ylabel('m',fontsize=16)
    plt.xlabel('h',fontsize=16)
    plt.ylim([-1,1])
    plt.xlim([-1,1])
    plt.show()

interactive(mfa_ising_h_vs_m, T=(0.1,5)) 

import plotly.graph_objects as go
import numpy as np
from scipy.optimize import fsolve
from scipy.optimize import root_scalar  # Importing root_scalar

def compute_xcross(T, h_over_T):
    
    def f(M):
        Tc = 1.0  # Normalized temperature
        h = h_over_T * T  # Compute actual h from h/T and T
        return M - np.tanh((M + h) / (T / Tc))
    
    # Using symmetric interval for root finding to allow negative solutions
    interval = [-2.0, 2.0]

    # Check if a root is likely within the given range
    if f(interval[0]) * f(interval[1]) > 0:
        return 0.0  # If no root likely, return 0
    
    # Find the root using bisection method within the specified range
    result = root_scalar(f, bracket=interval, method='bisect')
    
    return result.root 

# Define ranges for h/T  and T/Tc (from 0.2 to 1.5)
h_over_T_values = np.linspace(-0.2, 0.2, 101)
T_over_Tc_values = np.linspace(0.1, 1.6, 101)

# Create a meshgrid for T/Tc and h/T
H_over_T, T_over_Tc = np.meshgrid(h_over_T_values, T_over_Tc_values)

# Initialize an array for magnetizations
Magnetizations = np.zeros_like(H_over_T)

# Compute M for each (h/T, T/Tc) pair in the meshgrid
for i, T in enumerate(T_over_Tc_values):
    for j, h in enumerate(h_over_T_values):
        
        Magnetizations[i, j] = compute_xcross(T, h)


# Create a 3D surface plot using Plotly
fig = go.Figure(data=[go.Surface(z=Magnetizations, x=H_over_T, y=T_over_Tc, 
                                 colorscale='RdBu', 
                                 contours={
                                     #'z': {'show': True, 'start': -1.0, 'end': 1.0, 'size': 0.1, 'color':'orange'},
                                     'x': {'show': True, 'start': -1, 'end': 1, 'size': 0.05, 'color':'black'},
                                     'y': {'show': True, 'start': 0.5, 'end': 1.5, 'size': 0.1, 'color':'black'}
                                 })])

# Update plot layout
fig.update_layout(
    title='3D Plot of Magnetization M vs normalized temperature and field; T/Tc and h/T',
    scene=dict(
        xaxis_title='h/T',
        yaxis_title='T/Tc',
        zaxis_title='M',
        aspectmode='manual',
        yaxis_autorange='reversed',  # Reverse the Y-axis
        aspectratio=dict(x=2, y=2, z=0.75),  # Adjust these values as needed
        camera=dict(
            eye=dict(x=-4, y=-2, z=2),  # Adjust camera "eye" position for better view
            up=dict(x=0, y=0, z=1)  # Ensures Z is up
        )
    ),
    autosize=False,
    width=900,
    height=900
)

# Display the figure with configuration options
fig.show()

len(h_over_T_values)//2+1

51

plt.plot(Magnetizations[:, len(h_over_T_values)//2-1], color='blue')
plt.plot(Magnetizations[:, len(h_over_T_values)//2+1], color='blue')

[<matplotlib.lines.Line2D at 0x7fc38589de80>]

Critical Exponents

A hallmark of second-order phase transitions (critical phenomena) is the emergence of universal power-law behavior near the critical temperature \( T_c \):

\[ m \sim |T - T_c|^{\beta} \]

\[ c \sim |T - T_c|^{-\alpha} \]

\[ \chi = \frac{\partial m}{\partial B} \sim |T - T_c|^{-\gamma} \]

Correlation lengths \( \xi \) diverge at the critical point, reflecting the emergence of long-range order:

\[ f(r = |j - k|) = \langle s_j s_k \rangle \sim r^{-d + 2 - \eta} e^{-r/\xi} \]

\[ \xi \sim |T - T_c|^{-\nu} \]

Mean-Field Critical Exponents

We can derive the mean-field value of the exponent \( \beta \) by expanding the hyperbolic tangent near the critical point:

\[ \tanh(x) \approx x - \frac{1}{3}x^3 + \dots \]

Applying this expansion to the self-consistent equation:

\[ m = \tanh(\beta J q m) \approx \beta J q m - \frac{1}{3}(\beta J q m)^3 \]

• The trivial solution \( m = 0 \) corresponds to the disordered phase, valid above \( T_c \).

• For \( T < T_c \), a non-zero magnetization emerges. Solving perturbatively gives:

\[ m \approx \sqrt{3} \cdot \frac{T}{T_c} \left(1 - \frac{T}{T_c} \right)^{1/2} + \dots \]

• Hence, the mean-field critical exponent for magnetization is:

\[ \beta_{\text{MFA}} = \frac{1}{2} \]

Landau theory

# Calculate the free energy for different temperatures
T = 5
a, b = -1, 3

phi = np.linspace(-1, 1, 100)
F   = a * phi**2 / 2 + b * phi**4 / 4

# Plot the free energy as a function of the order parameter for different temperatures
plt.figure(figsize=(10, 6))
plt.plot(phi, F )
plt.xlabel('Order parameter')
plt.ylabel('Free energy')
plt.show()

Problems

1. Use Transfer matrix method to solve general 1D Ising model with \(h = 0\) (Do not simply copy the solution by setting h=0 but repeat the derivation :)

2. Plot temperature dependence of heat capacity and free energy as a function for \((h=\neq, J\neq 0)\) \((h=0, J\neq 0)\) and \((h=\neq, J\neq \neq)\) cases of 1D Ising model. Coment on the observed behaviours.

3. Explain in 1-2 sentences:

   • why heat capacity and magnetic susceptibility diverge at critical temperatures.

   • why correlation functions diverge at a critical temperature

   • why are there universal critical exponents.

   • why the dimensionality and range of intereactions matters for existance and nature of phase transitions.

   • why MFT predictions fail for low dimensionsal systems but consistently get better with higher dimensions?

4. Using mean field approximation show that near critical temperature magnetization per spin goes as \(m\sim (T_c-T)^{\beta}\) (critical exponent not to nbe confused with inverse \(k_B T\)) and find the value of \beta. Do the same for magnetic susceptibility \(\chi \sim (T-T_c)^{-\gamma}\) and find the value of \(\gamma\)

5. Consider a 1D model given by the Hamiltonian:

\[H = -J\sum^{N}_{i=1} s_i s_{i+1} + D\sum^{N}_{i=1} s^2_i \]

where \(J>1\), \(D>1\) and \(s_i =-1,0,+1\)

• Assuming periodic boundary codnitions calcualte eigenvalues of the transfer matrix

• Obtain expresions for internal energy, entropy and free energy

• What is the ground state of this model (T=0) as a function of \(d=D/J\) Obtain asymptotic form of the eigenvalues of the transfer matrix in the limit \(T\rightarrow 0\) in the characteristic regimes of d (e.g consider differnet extereme cases)